Consumption Inequality and Intra-Household
Allocations∗
Jeremy Lise
Department of Economics
Queen’s University
lisej@qed.econ.queensu.ca
Shannon Seitz
Department of Economics
Queen’s University
seitz@post.queensu.ca
December 1, 2004
Abstract
The consumption literature uses adult equivalence scales to measure individual level inequality. This practice imposes the assumption that there is no
within household inequality. In this paper, we show that ignoring consumption
inequality within households produces misleading estimates of inequality along
two dimensions. First, the use of adult equivalence scales underestimates the
level of cross sectional consumption inequality by 30%. This result is driven by
the fact that large differences in the earnings of husbands and wives translate
into large differences in consumption allocations within households. Second,
the rise in inequality since the 1970s is overstated by two-thirds: within household inequality declined over time as the share of income provided by wives
increased. Our findings also indicate that increases in marital sorting on wages
and hours worked can simultaneously explain virtually all of the decline in
within household inequality and a substantial fraction of the rise in between
household inequality for one and two adult households in the UK since the
1970s.
JEL Classification: D12, D13, D63, J12, J22
Keywords: Collective Model, Consumption Inequality, Marital Sorting, Adult
Equivalence Scales
∗
Department of Economics, Dunning Hall, Queen’s University, Kingston, Ontario, Canada, K7L
3N6. We would like to thank Garry Barrett, Richard Blundell, Martin Browning, Chris Ferrall,
Bernard Fortin, Allen Head, Bev Lapham, Arthur Lewbel, Frank Lewis, Hao Li, Huw Lloyd-Ellis,
Mary MacKinnon, Krishna Pendakur, José-Vı́ctor Rı́os-Rull, and participants at the Zeuthen Workshop at the University of Copenhagen, HEC Montreal, Queen’s University, Simon Fraser University,
Australia National University, the 2004 SOLE Annual Meetings, the 2004 CEA Annual Meetings,
Workshop on Consumer Behavior and Welfare, and the PhD conference in Business and Economics
in Perth, Australia for many helpful comments.
1
Introduction
A recent literature has documented a large rise in consumption inequality in several
developed countries.1 Underlying these measures of inequality is the use of adult
equivalence scales, which are used to assign consumption levels to each member of a
household. This has been necessary as there do not exist comprehensive measures of
individual level consumption for households with more than one member. However,
the drawback of this approach is that it implicitly assumes that there is no inequality
among adults within the household. In particular, the use of adult equivalence scales
implies a very restrictive model of the household in which husbands and wives split
consumption equally, regardless of the source of the income.2
This equal division assumption is inappropriate in the study of consumption inequality, as a large literature routinely rejects the assumption that the consumption
allocation does not vary with the source of income in the household (Blundell et al.,
2002; Browning, Bourguignon, et al., 1994; Browning and Chiappori, 1998; Chiappori
1988, 1992; Chiappori et al., 2002; Donni, 2001; Lundberg, Pollak and Wales, 1997;
Manser and Brown 1980; McElroy and Horney, 1981, among many others). Since
there has been a sizable increase in women’s wages and labor supply over the last half
century, the share of household earnings provided by the wife has changed substantially over time. If consumption allocations depend on the source of income and the
sources of income within households have changed over time, then adult equivalence
scales will produce an inaccurate picture of the trends in consumption inequality.
The goal of this paper is to document the trends in consumption inequality once
within household inequality is taken into account. We construct and estimate a static
1
See Blundell and Preston (1998) for the UK, Pendakur (1998) for Canada and Barrett, Crossley
and Worswick (2000) for Australia. The evidence on consumption inequality in the United States
is mixed. Slesnick (2001) and Krueger and Perri (2003) do not find a large rise in consumption
inequality since the 1970s. Alternatively, Cutler and Katz (1992) and Johnson and Shipp (1997)
report a rise in inequality. Attanasio, Battistin, and Ichimura (2004) use two sources of data on
consumption and also find consumption inequality was rising over time.
2
Recently, there has been a substantial departure from this literature. Browning, Chiappori,
and Lewbel (2004) relax the assumption that household members split consumption equally in the
construction of adult equivalence scales. Hong and Rı́os-Rull (2004) use information on the purchase
of life insurance to estimate equivalence scales.
model of intra-household allocations to examine how changes in the source of income
in the household translate into changes in individual-level consumption allocations.
In particular, we observe variation in labor supply, total household consumption, and
wages for singles and married couples in the data. We use the joint relationships
between consumption, wages and labor supply for single households and between
own and spousal wages in married households to infer the private consumption levels
for married men and women. The model is estimated on a sample of one and two
person households from the UK Family Expenditure Survey (FES) for the years 1968
to 2001. Under relatively weak identification assumptions, the model allows us to
infer the level of consumption allocated to each member of the household, which is
necessary for the measurement of individual level consumption inequality.
We then use our estimates to construct a new measure of consumption inequality
across individuals. We have two main findings. First, measures of consumption inequality that ignore the potential for intra-household inequality may underestimate
individual-level inequality by 30%, as the difference in earnings across husbands and
wives generates substantial within household inequality. Second, the rise in consumption inequality reported in the literature is overstated by two-thirds. This result is
due to the fact that within household inequality has fallen over time as female wages
and labor supply have increased. An implication of our findings is that the equal
sharing assumption implicit in adult equivalence scales is valid only for households in
which the wife has the same earnings as her husband.
In this paper, we provide an alternative picture of the trends in inequality. Our
analysis also sheds new light on the forces underlying the trends. We provide evidence on the importance of several potential explanations for the rise in consumption
inequality between households and the fall in inequality within households since the
1970s in the UK. While changes in the demographic composition of the population
appear to play a limited role, an increase in marital sorting has profound effects on
the trends in consumption inequality.3 In particular, the rise in marital sorting ob3
In this instance, the degree of marital sorting is measured by the correlation between character-
2
served in the data has the potential to account for all of the fall in within household
inequality and at the same time can explain a large fraction of the rise in consumption
inequality between households. Clearly, looking inside the household is as important
as looking between households for the study of consumption inequality.
The remainder of the paper is organized as follows. Section 2 describes in detail the
stylized facts on earnings and consumption inequality, wages, and labor supply that
provide the motivation for our study. Section 3 outlines the theoretical framework
and the identification strategy for estimating the rule to allocate consumption to
individuals within a household. Section 4 describes the data set and the strategy for
estimating the model. The estimation results are presented in Section 5. Section 6
presents a decomposition of consumption inequality and considers the importance of
several explanations for the trends in consumption inequality. Section 7 concludes.
2
Trends in Consumption and Earnings Inequality
in the UK
In this section, we outline the main stylized facts regarding consumption and income
inequality in the UK between 1968 and 2001. The data we use to conduct our analysis
comes from the UK Family Expenditure Survey (FES). The FES contains information
on household consumption expenditures and earnings over the period 1968 to the
present. In the construction of the following stylized facts, we restrict the sample to
individuals between the ages of 16 and 65 and eliminate students, retirees and the
self-employed. We are particularly interested in the following four features of the
data:
1. There has been a large rise in earnings inequality between individuals. Figure 1
documents the trend in the Gini index for the distribution of individual and
household earnings. The Gini index for individual earnings has risen by 12%
over the past 30 years. This rise in earnings inequality in the UK has been well
documented in the literature (e.g. Blundell and Preston, 1998).
istics, such as education and wages, across spouses.
3
2. Although earnings inequality between individuals is much higher than earnings
inequality between households, the latter rose much more rapidly: the Gini
index for inequality between households rose by 41% between 1968 and 2001.
3. As reported by Blundell and Preston (1998), there has been a corresponding
rise in consumption inequality. To account for economies of scale, we construct
a standard measure of individual-level consumption by dividing total household
consumption by the square root of household size. The Gini index for this
measure of consumption is presented Figure 1. The level of income inequality
is higher than the level of consumption inequality but the rise in inequality is
higher for consumption than for earnings.
4. As illustrated in Figure 2, the correlation between the earnings of husbands and
wives increased dramatically over time. This is due to both the fall in the gender
wage gap and the rise in female labor supply. Figure 3 highlights the dramatic
change in the gender wage gap and in women’s contribution to household labor
income between 1968 and the present. The dashed line represents the female’s
share of potential income, defined as the share of labor earnings that would be
contributed by the wife if both spouses worked full-time.4 The solid line represents women’s share of actual household earnings. Overall, potential earnings
of wives increased by 13.5%, and women’s share of earnings in the household
increased by 93% over the sample period. The latter partly reflects the increase
in women’s wages relative to those of men, but also the large changes in female
and male employment rates and hours worked since the 1960s.
In summary, the evidence presented here highlights the fact that there has been
a large rise in earnings and consumption inequality between households while at the
same time there has been a fall in inequality in the earnings distribution within
households.
4
For households with missing wage data due to non-participation, we include a predicted wage
based on a standard selection-corrected wage equation. Results are available upon request.
4
3
Theoretical Framework
In this section, we present a model that allows us to infer the level of consumption
for each member of a household. The model is based on Chiappori’s (1988, 1992)
collective model of household decision making.5 This framework is less restrictive
than the model of equal division underlying adult equivalence scales, as the only
restriction on the intra-household allocation process is that households reach Pareto
efficient allocations. We start with a description of the problem faced by single agents.
We then describe the intra-household allocation decision of married couples. Finally,
we outline the model restrictions that allow for the identification of the share of
consumption allocated to each household member.
3.1
Single Agents
Assume all single individuals have preferences over leisure and consumption. Denote
leisure, expenditures on private consumption and expenditures on public consumption
for an agent of gender g, g ∈ {m, f } by Lg , C g , and P , respectively.6 Labor supply is
denoted lg and the total time available to agents is normalized to one, i.e. lg = 1−Lg .
Denote total household non-labor income net of savings by Y . Labor earnings are
denoted wg (l) and include any after tax income that depends directly on the labor
supply decision. In particular, wg (l) includes unemployment insurance benefits paid
to individuals who are not working.7 Preferences for single agents are described by
U g (ug (Lg , C g ), P ), where it is assumed preferences over private consumption goods
and leisure are separable from preferences over public consumption goods. Single
agents choose labor supply and consumption to maximize utility, subject to the budget
constraint:
max U g (ug (Lg , C g ), P )
Lg ,C g ,CP
5
See also Browning et al. (1994), Browning and Chiappori (1998), Chiappori et al. (2002) and
Blundell et al. (2002)
6
Chiappori, Blundell and Meghir (2002) establish conditions under which the collective model
with public goods is identified.
7
We construct labor earnings in this fashion, as unemployment benefits are paid directly to one
person in the household and likely affect allocations differently than does shared non-labor income.
5
subject to C g + P = wg (l) + Y.
3.2
Married Couples
Consider a two member household, where each member has distinct preferences over
own leisure, own private consumption, and household public consumption. Denote
by C a Hicksian composite good that contains private and public consumption:
C = C f + C m + P.
As with singles, we assume that private consumption and leisure (C g , Lg ) are separable from consumption of the public good (P ) for married couples. Preferences for
a married person of gender g can be described by:
V g (v g (Lg , C g ), P ),
where v g (Lg , C g ) captures preferences over private consumption and leisure. Under
the assumptions that preferences are egoistic and that allocations are Pareto efficient,
the household’s allocations are the solution to the problem:
max
Lf ,Lm ,C f ,C m ,P
λV f (v f (Lf , C f ), P ) + (1 − λ)V m (v m (Lm , C m ), P )
(1)
subject to C f + C m + P = wf (l) + wm (l) + Y.
The Pareto weight, λ, represents the female’s bargaining power within the household,
and will typically be a function of full-time labor income (wf (1), wm (1)), non-labor
income (Y ) and other “distribution factors” (z) that influence household bargaining
power, but do not have an effect on individual preferences, as in Chiappori, Fortin,
and Lacroix (2002).
Chiappori (1992) shows that the intra-household allocation problem faced by a
husband and wife can be decentralized by considering a two stage process. In the first
stage the husband and wife decide on the level of public good consumption (P ) and
on how to divide the remaining non labor income y = Y − P . The assumption that
consumption of the public good is separable from leisure and private consumption is
6
key to allowing the allocation of public consumption to occur in the first stage (see
Chiappori, Blundell, and Meghir (2002) for details).8 Define the sharing rule φ(y, z)
as the amount of non-labor income that is assigned to the wife. Then y − φ(y, z) is
non-labor income assigned to the husband.
In the second step, each household member chooses his or her own private consumption and leisure, conditional on the level of public consumption and the budget
constraint determined in the first stage:
max v g (Lg , C g )
Lg ,C g
(2)
subject to C g = wg (l) + φg (y, z),
where φf = φ(y, z) and φm = y − φ(y, z). The Pareto problem represented in (1) and
the sharing rule interpretation in (2) produce identical labor supplies and consumption
demands, under the assumption that an efficient level of public consumption is chosen
in the first stage.
3.3
Identification of the Sharing Rule in the Case of Quadratic
Preferences
The question we aim to address in this paper is whether measures of consumption
inequality from the collective model differ from measures in the literature based on
standard equivalence scales. To provide an answer to this question, it is necessary to
obtain an estimate of the full sharing rule to uncover the share of income allocated
to each household member for consumption. In this case, the first order conditions
of the sharing rule are not sufficient for identification. We therefore need to impose
an additional restriction on preferences. As in Vermeulen (2003) and Browning,
Chiappori, and Lewbel (2004), we assume that married individuals have the same
marginal utility from private consumption as single individuals, but possibly different
marginal utility for leisure and public consumption. In Section 5, we test whether
this assumption is supported by the data.
8
We do not incorporate savings in the model. However, under the assumptions that preferences
are time separable and the level of savings is efficient, we can obtain estimates of the sharing rule
while abstracting from the savings decision of the household.
7
Our treatment of households extends the models of Blundell, Chiappori, Magnac,
and Meghir (2002) and Vermeulen (2003) to allow for households in which both
spouses do not necessarily work full time and in which both spouses make labor force
participation decisions.9 In particular, we assume that individuals can choose from H
discrete labor supply possibilities, in addition to non-participation.10 Further assume
that Lf , Lm , Y , wf (l), and wm (l) are observed in the data. As is consistent with
our empirical exercise, C and P are observed although the distribution of private
consumption between the husband and wife (C f and C m ) is not observed.
Let preferences for private consumption and leisure be represented by a quadratic
direct utility function, a flexible form representing a second-order Taylor series expansion in leisure and consumption. The utility a single individual of gender g derives
from labor supply choice h is:
Uhg = v g (lh , Chg ) + ω g (P ) + εgh
g
g
= αlg lh + αllg lh2 + αcl
lh Chg + αcg Chg + αcc
(Chg )2 + αPg Phg + αPg P (Phg )2 + εgh ,
and the utility a married individual of gender g derives from labor supply choice h is:
g
g
Vhg = βlg lh + βllg lh2 + αcl
lh Chg + αcg Chg + αcc
(Chg )2 + βPg Phg + βPg P (Phg )2 + εgh ,
where εgh is an unobserved preference component that is assumed to be distributed iid
across individuals and labor supply choices.11 This specification allows the marginal
utility of leisure and public consumption to differ between married and single men
and women, but restricts the marginal utility of private consumption to be the same
for both married couples and singles.
9
Blundell et al. (2002) model the labor force decision of the wife as continuous and of the husband
as discrete; either he works full time or not at all. Vermeulen (2003) considers the case where males
are assumed to work full-time and females face a discrete labor supply choice which includes the
option of non-participation.
10
This assumption is not necessary for identification, and is not very restrictive, as the discrete
choice of hours can be any integer value of weekly hours.
11
We relax this assumption in Section 5.2.
8
Assume the sharing rule is linear in the distribution factors:
Ã
!
K
X
φ(y, z) =
φ0 +
φk zk y
k=1
>
= (z φ)y,
where there are K distribution factors plus a constant in the vector z and where
y is non-labor income net of expenditures on the public good. We can condition on
household expenditures on the public good for both singles and married couples under
the assumptions that households make efficient decisions in the first stage and that
preferences over public goods are separable from preferences over consumption and
leisure (Deaton and Muelbauer, 1980).
The budget constraints for the second stage of the budgeting process can be expressed as:
Chg = wg (lh ) + y
(3)
Chf = wf (lh ) + (z> φ)y
(4)
Chm = wm (lh ) + (1 − z> φ)y
(5)
for single individuals,
for married women and
for married men.
Only differences in utility between labor supply choices matter in the model; thus
the parameters must be estimated relative to a base case. We assume that the choice
of not working (h = 0) is the base case. After substituting the budget constraint into
the utility function, the difference between working h > 0, ∀h ∈ {1, 2, ..., H} and not
working (h = 0) for single men and women can be expressed as:
£ g
¤
g
g
Uhg −U0g = αlg lh +αllg lh2 +αcl
lh · w̃g (lh )+αcg w̃g (lh )+αcc
[w̃ (lh )]2 + 2w̃g (lh ) · y +εgh −εg0 ,
(6)
where w̃g (lh ) = wg (lh ) − wg (l0 ) and [w̃g (lh )]2 = [wg (lh )]2 − [wg (l0 )]2 . Consider next
the problem of a married woman. The difference between working h > 0, ∀h ∈
9
{1, 2, ..., H} and not working (h = 0) is described by:
f
f
Vhf − V0f = βlf lh + βllf lh2 + αcl
lh · w̃f (lh ) + αcl
(z> φ) · lh · y
(7)
f
f
+αcf w̃f (lh ) + αcc
[w̃f (lh )]2 + 2αcc
(z> φ) · w̃f (lh ) · y + εfh − εf0 .
Finally, consider the problem of a married man, where the difference between working
h > 0, ∀h ∈ {1, 2, ..., H} and not working (h = 0) is described by:
m
m
m >
Vhm − V0m = βlm lh + βllm lh2 + αcl
lh · w̃m (lh ) + αcl
lh · y − αcl
(z φ) · lh · y + αcm w̃m (lh )
m
m >
m
m m
(z φ) · w̃m (lh ) · y + εm
+αcc
[w̃m (lh )]2 + 2αcc
w̃ (lh ) · y − 2αcc
h − ε0 . (8)
The parameters βlg , βllg are directly identified from the data on married individuals.
g
g
Given αcl
, αcg , and αcc
, identified from data on single individuals, it is straightfor-
ward to recover the sharing rule parameters (φ).12 It is the sharing rule parameters
that allow us to determine the level of consumption enjoyed by each member of the
household.
The differences in utility described by Equations (6), (7) and (8) can be expressed
for single individuals in reduced form as:
Uhg − U0g = Πgl lh + Πgll lh2 + Πgly lh y + Πglwl lh w̃g (lh ) + Πgwl w̃g (lh ) + Πg(wl)2 [w̃g (lh )]2
+Πgwly w̃g (lh ) · y + εgh − εg0 ,
and for married individuals as:
Vhg − V0g = Πgl lh + Πgll lh2 + Πgly lh y + Πglwl lh w̃g (lh ) + Πgwl w̃g (lh ) + Πg(wl)2 [w̃g (lh )]2
+Πgwly w̃g (lh ) · y + Πglym lh y + Πgzlym · zlh y + Πgwlym w̃g (lh ) · y
+Πgzwlym · zw̃g (lh ) · y + εgh − εg0 ,
where the Πs are reduced form parameters. The system above implies a set of overidentifying restrictions for the sharing rule parameters that enable us to jointly test the
g
g
g
g
The parameters capturing preferences over the public consumption good (αP
, αP
P , βP , βP P )
can not be identified as the utility from consumption of the public good is the same regardless of
the labor supply decision. One implication is that we will be able to estimate the sharing rule but
not fully recover preferences. As a result, we cannot make welfare comparisons.
12
10
assumptions of the collective model, the functional form for preferences, the sharing
rule, and the assumption that the marginal utility of private consumption is the same
regardless of marital status:
φ0 =
φk =
Πflym
Πfly
+1=
Πfzk lym
Πfly
=
Πfwlym
Πfwly
Πfzk wlym
Πfwly
Πm
Πm
lym
wlym
+1=− m =− m ,
Πly
Πwly
=−
Πm
Πm
zk wlym
zk lym
=
−
,
m
Πm
Π
ly
wly
(9)
k = 1 . . . K.
In the following section, we outline our strategy for estimating the model and testing
the above restrictions using consumption data from the UK.
4
Empirical Specification
4.1
Data
The data we use comes from the UK Family Expenditure Survey (FES). This data
is ideal for the study of consumption inequality for three reasons. First, it contains
detailed information on private and public consumption expenditures for households,
on wages and labor supply for individuals, and on demographics including age, sex,
education (from 1978 onward) and region of residence. Second, the FES has fewer
problems with measurement issues than the leading contenders in the US and elsewhere.13 The FES uses a weekly diary to collect data on frequently purchased items
and uses recall questions to collect data on large and infrequent expenditures. Finally,
the FES contains information over the period 1968 to the present, which allows the
study of changes in consumption inequality over a long period of time.
Our sample contains single person households and couples without children.14 We
restrict the age range in the sample to individuals between the ages of 22 and 65 and
eliminate students and the self-employed. Households in which one of the individuals
13
Battistin (2003) documents reporting errors in the US Consumer Expenditure Survey due to
survey design.
14
We exclude households with children in this paper to abstract from the intra-household allocation
of resources for children’s consumption. This is obviously an important issue. To this end, our
estimates of the sharing rule and the comparison of various inequality measures only apply to
households without children. We leave to future work an analysis of consumption inequality for the
entire sample of households.
11
is in the top one per cent of the wage distribution are also excluded. The resulting
sample contains 87, 668 individuals.15 Descriptive statistics for our entire sample are
presented in Table 1.
We define consumption and non-labor income measures as follows. Total consumption is defined as total household expenditures. Public consumption is defined
as expenditures on housing, light and power, and household durable goods. Private
household consumption is total expenditures net of public consumption. Other income is defined as total household expenditures minus net labor income. We use this
expenditure based definition of non-labor income, as it is consistent with the assumptions of a two stage budgeting process, time separable preferences, and separability
of public goods consumption from leisure and private consumption as in the model.16
To construct the level of consumption corresponding to each labor supply decision,
including zero hours, we need to assign an earnings level to all individuals. For those
who are working we use the usual hourly wage, defined as weekly earnings divided by
usual weekly hours. For non-participants we use a predicted wage, computed based
on a reduced form selection-corrected wage equation.17 After tax earnings are subsequently computed by converting weekly wage income to an annual base, deducting the
appropriate personal allowance and then applying the appropriate tax rate. Personal
allowances and marginal tax rates are from the Board of Inland Revenue (1968–2001).
All monetary values are expressed in 1987 pounds. The resulting income measure is
treated as known and is used to construct the within household distribution factor
defined as the potential share of household labor income contributed by the wife,
z1 = wf /(wf + wm ). Individuals may also be entitled to income related to earnings
15
The sample size in 1968 is 2,584 and the sample size in 2001 is 2,757. The sample sizes do not
vary markedly across years: the smallest sample is 2,502 in 1979 and the largest is 2,932 in 2000.
16
In estimation, household expenditures on public goods are subtracted from other income, resulting in non-labor income net of public goods consumption. In addition to the separability assumptions, wage profiles are assumed to be exogenous. This rules out the possibility of job-specific
human capital accumulation.
17
The log of the wage is estimated as a function of age, birth cohort, year, quarter, and regional
dummies, plus the age at which full time education was completed, and its square. The selection
equation is identified by the exclusion from the wage equation of household non-labor income, marital
status, and the age, education, and the labor income of the spouse. Results are available upon
request.
12
when working zero hours, for instance unemployment benefits, so we also predict unemployment benefits for those who are working based on the Official Yearbook of the
United Kingdom (1968-2001).
Labor supply is measured by a discrete variable that takes on three values: not
participating, working part-time and working full-time. Full time is defined as working
35 hours per week or more, and part-time is defined as 1 to 34 hours per week. The
choice of these ranges is based on the hours histograms in Figure 4, which suggests
a full-time definition of 35 hours a week or more. The average hours worked in the
part-time category is approximately 20 hours per week, and approximately 40 hours
per week in the full-time category.
In order to ensure consistency between the number of hours worked in each of
the three states and the corresponding consumption level we adopt the following
convention. If an individual is observed to be working either part-time or full-time
we use the reported number of hours to measure labor supply and usual take home
pay in constructing the consumption. In cases for which we do not observe the labor
supply state, we calculate after tax earnings based on 20 hours for the part-time choice
and 40 hours for the full-time choice. Constructing individual consumption in this
way ensures our measure of total private consumption in the household is consistent
with that observed in the data.
Likely candidates for the distribution factors are the wife’s potential share of total
household labor income (wif /(wif + wim )), the local sex ratio (Seitz, 2004), and an
index of the generosity to the wife of local divorce legislation (Chiappori, Fortin, and
Lacroix, 2002). At present, we consider the wife’s share of potential labor earnings,
presented in Figure 3, and the age gap between spouses as distribution factors in
estimation.
4.2
Econometric Specification
The model of Section 3.3 can be estimated using a multinomial logit under the assumption that the disturbances εih are independent and identically distributed type
13
I extreme value. Let dgih denote an indicator equal to 1 if individual i makes labor
supply choice h and zero otherwise. The contribution of individual i to the likelihood
function is the probability of observing individual i making labor force decision h,
which has the form:
Pr(dgih = 1) = Pr(ugih > ugij , ∀j 6= h; j, h ∈ {0, 1, ..., H})
exp(v g (Lih , Cih ; Xi , zi ))
= PH
j=0
exp(v g (Lij , Cij ; Xi , zi ))
.
In estimation, heterogeneity in preferences for leisure is introduced through the
vector X which includes age, birth cohort, education, marital status, region, and
quarter and year to control for seasonality and cyclical effects.18 The parameters Πgl
and Πgll are assumed to be linear functions of the observed characteristics so that,
with a slight abuse of notation, for individual i we have
Πgl = Xi Π̃gl
Πgll = Xi Π̃gll .
Estimation proceeds in two steps. First, we estimate a selection-corrected wage
equation and predict wages for individuals that are not working. Second, we estimate
the discrete labor supply choice, treating wages as known.
5
Estimation Results
We begin with estimates of the sharing rule parameters, the parameters that allow
us to infer the share of consumption attributed to each adult in the household. As
discussed in Section 3.3, with quadratic utility and under the assumption that the
marginal utility of private consumption is the same for married and single individuals, we can construct each of the sharing rule parameters in four different ways from
estimates of the reduced form. The sharing rule parameters recovered from the reduced form estimates for two specifications of the model are presented in Table 2.
18
In order to break the collinearity between age, birth cohort and year we follow Deaton (1997)
and transform the year dummy variables so that the coefficients are orthogonal to a time trend and
sum to zero over the period 1968 to 2001.
14
The first column of the table presents estimation results from the case in which the
only distribution factor is the share of women’s potential earnings in household potential earnings. The second column presents results from a model where a second
distribution factor, the age difference between spouses, is included.
The estimated sharing rule parameters constructed from the different model restrictions described by Equation 9 are qualitatively similar, remarkably so for men. In
both specifications, the positive sign on φ1 indicates that an increase in the female’s
share of potential earnings increases her share of total consumption in the household.
The negative sign on φ2 suggests that the share of consumption women receive is
decreasing in the relative age of their husbands. The sharing rule parameters for
the second set of restrictions in Equation 9 are larger in absolute value for both the
intercept and the distribution factors. Upon closer examination of the reduced form
results, we find the reason for this difference across the estimates is due primarily to
the fact that the estimated value of the denominator, Πfwly , is relatively small. This
parameter captures the effect of the interaction between non-labor income and earnings for women on the labor supply decision. Since many women are not working,
we need to impute earnings for 39% of the women in the data. Most of the information used to predict wages is also included directly in the reduced form model for
hours; as a result, the predicted wage includes very little information. As a result,
the parameter estimate is likely biased towards zero. It should be noted that this set
of restrictions is less precise; as a result, it has less weight in the minimum distance
estimation used to obtain the sharing rule estimates as discussed below.19
The test statistics associated with several tests of the model restrictions are presented in the bottom five rows of Table 2. A Wald test on the model with one
distribution factor rejects the full set of restrictions. The Wald test on the full set
of restrictions from the model with two distribution factors, however, suggests the
model is not strongly rejected. We subsequently test whether the sharing rule param19
For comparison purposes, we also estimated a version of the model where predicted wages were
used in place of actual wages for all individuals and found no substantial changes in the parameter
estimates. Results are available from the authors upon request.
15
eters estimated from the restrictions within gender are the same. The test statistics,
presented in Rows 2 and 3 of the bottom panel of Table 2, indicate the within gender
restrictions are not rejected by the data. We also test whether each of the individual
restrictions from the female’s problem are consistent with the corresponding restrictions from the male’s problem in Rows 4 and 5. In each case, the test statistics indicate
the model restrictions are not rejected at conventional significance levels. Overall, the
test statistics provide some support for our version of the collective model.
We next compute consistent estimates of the sharing rule parameters from the
unrestricted estimates by minimizing the distance between the reduced form and
structural parameters, using the estimated covariance matrix from the reduced form
to construct the weighting matrix. The results of this exercise are presented in Table 3.20 The estimates suggest that a 10% increase in the share of potential earnings
attributed to the wife results in a 16% increase in the share of non-labor income she
receives. This result is consistent with an increase in the wife’s threat point within
a bargaining model. The estimate of φ2 indicates that an increase in the husband’s
age by 1 year results in a 0.4% decrease in the wife’s share of non-labor income.
While small in magnitude, this finding suggests that older spouses tend to have more
bargaining power in marriage.
5.1
Adult Equivalence Scales Revisited
One of the main goals of this paper is to determine whether measures of consumption
inequality using standard adult equivalence scales provide an accurate estimate of
consumption inequality across individuals. Recall, adult equivalence scales assume
that husbands and wives share in household consumption equally. In this section we
determine the conditions under which our model would yield the same measures of
consumption inequality as measures using adult equivalence scales. We set the age
difference between spouses to the average age difference in the data. We then use
the sharing rule estimates to determine what value of the female’s share in potential
20
See the Appendix for further details on the minimum distance estimation exercise.
16
household earnings satisfies:
wf
1
= φˆ0 + φˆ1 · f
+ φˆ2 · (2.09).
2
w + wm
Using estimates for φ0 , φ1 , and φ2 of −0.31, 1.59, and −0.004 respectively yields
51%. In other words, the model predicts that consumption is split equally across the
husband and wife when they have the same earnings!21 It is worth emphasizing that
this result is derived not from a model in which equal sharing is assumed: the only
assumptions imposed in estimation are that households make Pareto efficient decisions, that public consumption is separable from private consumption, and that the
marginal utility from private consumption is the same when single as when married.
5.2
Sensitivity Analysis
In this section, we consider the sensitivity of our results to both the sample used to
estimate the model and the robustness of our estimation results to several modifications. First, we consider whether the trend in inequality measured in the literature
for the entire population differs from the trend for our sample of one- and two-person
households. Upon examination of Figure 5, we see that the trends in between household inequality using adult equivalence scales are virtually the same across samples.
Thus, it does not appear that our results are driven exclusively by trends in inequality that are unique to our sample. That said, we are not able to say how the trends
in within household consumption inequality vary between our sample and the entire
population without estimating an extended version of our model.
With regards to robustness of our estimated results, the first check we consider is
whether the results are sensitive to our definitions of public and private consumption.
We first estimate the model under the assumption that there are no public goods and
then sequentially add housing, heat and lighting, household durables, transport and
services to public good consumption.22 The results of this exercise are reported in
21
To be precise, husbands and wives will split consumption equally when they have approximately
the same wages and hours.
22
Full estimation results are available from the authors upon request.
17
Table 4. With the exception of the zero public goods case, the parameter estimates
are quite robust across specifications: an increase in the wife’s share of potential
household earnings of 10% results in an increase in her consumption share of between
12% and 17%. Under the most restrictive assumption, that no goods are public in the
household, the model predicts women receive 40% of the consumption in households
where both spouses choose the same hours of work and have the same wage. As the
fraction of public goods in household expenditures increases, women receive a greater
share of consumption. This result reflects, in part, the fact that a larger portion of
consumption in the household is public and is thus split equally across spouses.
The next specification we estimate allows for differences in the sharing rule parameters for each birth cohort in our pooled sample. The sample covers a long time
period and a wide age range in every year; we thus estimate sharing rules for each
ten-year cohort in the data. The parameter estimates are presented in Columns 1
and 2 in Table 5 for the models with one distribution factor and two distribution
factors, respectively. With the exception of the 1900 and 1960 birth cohorts (which
have relatively small samples), the parameter estimates and the predicted share of
consumption assigned to wives when earnings are equal across spouses are quite similar across the cohorts. For the cohorts between 1910 and 1950, an increase in the
wife’s share of potential household earnings increases her share of non-labor income
between 13% and 23% and the estimated effect of an increase in the husband’s age
by one year fall within the range of −0.6% and 0.5%. The fact that the sharing rule
parameter estimates are quite similar across specifications is surprising considering
the possibility of large changes in divorce costs and gains to marriage over time.
The final robustness check we perform is to add unobserved heterogeneity in preferences to the model. We do this for two reasons. First, we want to allow for the
possibility that the preference shocks are correlated across labor supply choices. Second, we want to allow for additional flexibility in estimating preferences over leisure.23
Results from this specification for the model with one distribution factor are presented
23
2
We specify Πl = Xi Π̃l + uhi and Πll = Xi Π̃ll + uhhi , with uhi ∼ N (0, σh2 ) and uhhi ∼ N (0, σhh
)
18
in Column 3 of Table 5. Incorporating unobserved preference heterogeneity appears
to reduce both φ0 and φ1 slightly but does not change the implications of the model.
In particular, the effect of a 10% increase in potential household earnings attributed
to wives results in an increased transfer of between 11% and 21% for the 1910 to 1950
cohorts, which is close to the range reported in Column 1 above.
6
Consumption Inequality
In this section, we compare the inequality measure implied by our model to a conventional measure of consumption inequality. For the purposes of this analysis, we use
estimates of the model with two distribution factors and no unobserved preference
heterogeneity to construct our benchmark sharing rule.24 We use this sharing rule
to divide non-labor income between the husband and wife in each household. We
subsequently construct private consumption based on the individuals’ share of nonlabor income and his or her personal net labor earnings, where private consumption
is constructed as in equations (4) and (5). Our sharing rule measure of individual
consumption, for married individuals, is then equal to individual private consumption
plus household public consumption:
C f = wf (lh ) + [φˆ0 + φˆ1
wf (1)
+ φˆ2 (agem − agef )] · y + P
wf (1) + wm (1)
and
C m = wm (lh ) + [1 − φˆ0 − φˆ1
wf (1)
− φˆ2 (agem − agef )] · y + P.
wf (1) + wm (1)
Single individuals consume their entire labor and non-labor income. For comparison
purposes, we construct another measure of individual consumption, equal division,
which assumes that all consumption is divided equally between the husband and wife.
(see Train (2003)). The contribution to the likelihood function then becomes
Z Z
exp(v g (Lih , Cih ; Xi , zi , uhi , uhhi ))
Pr(dgih = 1) =
dF (uhi )dF (uhhi ),
PH
g
j=0 exp(v (Lij , cij ; Xi , zi , uhi , uhhi ))
which does not have a closed form solution, but can be estimated using Simulated Maximum Likelihood.
24
See Column 2 of Table 3.
19
In the equal division case, individual consumption is calculated as household public
consumption plus one half of household private consumption. In both the sharing
rule and the equal division case, we double count public consumption. This accomplishes the same end as using an equivalence scale to assign household consumption
to individual members.25
Having constructed these two measures of individual consumption, we can construct a time series of inequality measures and decompose them into changes in between and within household inequality. While the Gini coefficient is a well known
and widely used inequality index, it does not allow overall inequality to be exactly
decomposed into within and between group contributions. As this is one of the main
objectives of this paper we also compute the Mean Logarithmic Deviation (MLD)26
in consumption, defined as
n
1X
Iα (C) =
log
n i=1
µ
µC
Cig
¶
,
(10)
where µC is the mean level of consumption. The index of total inequality using the
MLD can be additively decomposed into within and between household inequality:
IαT (C) = IαW (C) + IαB (C),
(11)
where IαW (C) is within household inequality and IαB (C) is between household inequality. Under the assumption of equal division, within household inequality is zero;
therefore, we can calculate IαB (C) by using equal division. Using individual consumption constructed with the sharing rule we obtain the total inequality index IαT (C).
We can then recover intra-household inequality using Equation (11).
6.1
Between and Within Household Consumption Inequality
The time-series trend of total and between household inequality for the years 1968
to 2001 is presented in Figure 6. The Gini index measures are presented in the first
25
It should be noted that the correlation between our equal division consumption inequality measure and a measure of inequality using equivalence scales is 0.99.
26
The MLD is a member of the Generalized Entropy Class, the only class of additively decomposable inequality indices (Shorrocks, 1984).
20
panel, and the MLD measures of inequality are presented in the second panel. Inequality was stable from 1968 to 1980 at which time it increased substantially until
around 1990, and has been falling slightly from 1990 through 2001. Of particular
interest are two findings. First, ignoring intra-household inequality underestimates
consumption inequality in 1968 substantially. This is due to the large differences
in earnings between husbands and wives in 1968, differences that translate into an
unequal distribution of consumption in the household. It is not surprising that our
measure of inequality in 1968 is higher than the conventional measure: by definition, taking within household inequality into account will increase total inequality.
However, the magnitude of the difference is striking: consumption inequality, as measured using adult equivalence scales, is underestimated by approximately 30% (15%)
when inequality is measured using the MLD (Gini index). To put this figure into
perspective, the difference between the level of inequality in 1968 using our measure
of inequality compared to the conventional measure is as large as the entire increase
in inequality between 1968 and 2001.
Second, the rise in consumption inequality under equal division, or between household inequality, may be over-stated by as much as 65%, as illustrated by the trend in
the MLD presented in Figure 7. The reason our measure of inequality is so different
from the equal division measure is due to the large fall in within-household inequality.
The stylized facts presented in Section 2 point to the two main reasons for the decline
in within household inequality: the fall in the gender wage gap and the rise in female
labor supply. As women’s wages rose and as married women increased their labor
supply, the share of income contributed to the household by the wife increased. The
share of consumption allocated to wives increased accordingly.
Consider the distribution of consumption inferred from our model for 1968 and
2001 in Figure 8. The dispersion in within household inequality decreased between
1968 and 2001. The fact that there was a high degree of dispersion in 1968 was a
major contributor to the high level of inequality at the beginning of the sample period.
Despite the fall in dispersion, the righthand panel of Figure 8 shows that there is still
21
a substantial deviation from equal division by 2001. Incorporating within household
inequality thus changes the picture of inequality dramatically.
6.2
Accounting for the trends in consumption inequality
In this section, we examine several explanations for the rise in consumption inequality
observed in the data. We focus on the years 1978 to 2001, as the major changes in
inequality occurred over the 1980s. Results are presented in Table 6. The first
two rows contain the benchmark inequality measures for 1978 and 2001, followed
by the absolute change over time in the third row. The rest of the table presents
the percentage of the change in the observed inequality measures attributed to each
explanation we consider.
6.2.1
Wages and Labor Supply
According to the stylized facts, two of the most salient trends over time are the
closing of the gender gap in wages and the rise in female labor supply. To what
extent do changes in the distribution of wages account for the rise in consumption
inequality? To answer this question, we conduct two exercises. First, we re-weight
the data for 1978 so that the wage distribution in 1978 matches that of 2001.27 This
experiment captures the fall in the gender wage gap and aggregate changes in the
wage distribution, but not changes in sorting on wages within households. In the
second experiment we re-weight the joint spousal distribution of wages to capture the
effect of sorting on inequality. Both experiments are subsequently repeated for labor
supply. The results are presented in Table 6.
What is interesting about the results on wage and hours sorting is that they can
simultaneously explain both the rise in consumption inequality across households and
the fall in consumption inequality within households: sorting on wages alone can explain approximately 39% of the rise in between household inequality and 78% of the
27
In particular, we construct histograms of log wages for 1978 and 2001. The histograms used to
re-weight the wage distributions have 10 bins each. We re-weight 1978 data so that the histograms
of log wages are the same in both years.
22
fall in within household inequality. With respect to sorting on hours, 32% of the rise
in between household inequality and 98% of the fall in within household inequality
can be explained by increased sorting within marriage. Regardless of the measure
of consumption inequality considered, the exercises conducted above illuminate the
dramatic role of sorting in determining the distribution of consumption across individuals.28
6.2.2
Demographics and Household Composition
Next, we consider the hypothesis that the rise in consumption inequality is capturing
cohort effects due to the changes in the age structure of the population. To this end,
we re-weight the 1978 data so that the age structure is the same as that in 2001,
holding all else constant. The results of this exercise, presented in the fifth row of the
bottom panel of Table 6, suggest that changes in the age distribution between 1978
and 2001 had virtually no effect on consumption inequality.
The second explanation we consider is the large change in household composition
that occurred alongside the rise in inequality. In particular, with delays in marriage
and a rise in divorce rates, the fraction of households with one adult increased relative to the fraction of two adult households.29 To assess the importance of changing
household composition, we re-weight the 1978 data so that the fraction of married
couples, the fraction of single women, and the fraction of single men in the population
match the proportions in 2001. The results of this exercise suggest that changes in
household composition can explain up to 17% of the change in household inequality
according to the sharing rule estimates and 16% of the change in consumption inequality when measured using adult equivalence scales. Together, a combination of
28
The compression of marginal tax rates also appear to have played a role in generating the sharp
rise in between household inequality during the 1980s. The top and bottom marginal tax rates are
plotted in Figure 9, where the top marginal rate falls from 83 per cent in 1978 to 60 per cent in
1979, and then falls again to 40 per cent in 1988. The increase in between household consumption
inequality is closely linked to the increase in after tax income inequality that occurred over the
1980s. It appears that changes in marginal tax rates had the effect of increasing between household
inequality substantially while having only a modest effect on within household inequality.
29
Although single person households have no within household inequality by definition, it is still
the case that there may exist substantial inequality between single adult households.
23
a changing age distribution and the change in household composition over time can
explain 16% of the change in the MLD over time, most of this effect coming through
household composition. In summary, our experiments indicate that the main driving
force behind the trends in consumption inequality are changes in marital sorting, not
changes in demographics. This finding reaffirms the fact that our understanding of
consumption inequality is incomplete when intra-household behavior is ignored.
7
Conclusion
The literature on consumption inequality has focussed on the question of how changes
in income inequality translate into changes in consumption inequality at the level of
the individual. In this paper, we show that current measures of consumption inequality only reflect inequality at the household level, as it is assumed there is no
inequality within households. We provide evidence that this assumption produces
a very inaccurate picture of both the level and trend in consumption inequality for
two reasons. First, the large dispersion in incomes in the household are highly inconsistent with equal division of consumption. When equal division is relaxed, there is
substantial inequality within the household. This suggests previous work underestimates the level of individual consumption inequality by 30%. Second, the earnings
gap within households has closed over the past 30 years, resulting in less inequality in the household over time. As a result, we estimate that the rise in inequality
documented in the literature is overstated by 65%. Our results show that the conventional approach for measuring inequality is only appropriate for measuring inequality
for single individuals and for couples with exactly the same earnings.
Our work highlights the fact that looking inside the household is as important as
looking between households for the study of consumption inequality across individuals. Ignoring what happens in the household not only changes our estimates of the
level and trend in inequality; it also changes our understanding of the forces behind
the trends. We find that increased marital sorting on earnings can explain both the
fall in within household inequality and the rise in between household inequality over
24
time. These results are complementary to those of Fernández and Rogerson (2001),
among others, on sorting and income inequality and suggest an important avenue for
further research.
25
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28
A
Minimum Distance Estimator of Structural Parameters
The structural parameters
³
´>
f
f
f
f
m
m
m
m
θ = φ0 , φ1 , φ2 , αcl
, αcc
, αcl
, αcc
, αlX
, αlX
, βlXm
, βlXm
can be consistently estimated by using a minimum distance estimator (MDE) (see
Chamberlain (1984)). We define the MDE as
³
³
´>
´
θ̂ = arg min Π̂ − f (θ) V −1 Π̂ − f (θ) ,
θ
where the function f imposes the structural restrictions on the reduced form, and V
is the covariance matrix of the reduced form parameter estimates. For the case in
which the sharing rule is a linear function of three distribution factors the structure
of the model implies the following restrictions on the reduced form parameters:



















³
´

Π̂ = f (θ) =⇒ 



















Π̂fly
Π̂flym
Π̂fz1 lym
Π̂fz2 lym
Π̂fwly
Π̂fwlym
Π̂fz1 wlym
Π̂fz2 wlym
Π̂m
ly
m
Π̂lym
Π̂m
z1 lym
Π̂m
z2 lym
Π̂m
wly
Π̂m
wlym
Π̂m
z1 wlym
Π̂m
z2 wlym
Π̂flX
Π̂m
lX
Π̂flXm
Π̂m
lXm

f
=
αcl
f
= αcl
(φ0 − 1) 

f

=
αcl
φ1


f

=
αcl
φ2

f

=
αcc

f
= αcc (φ0 − 1) 


f
=
αcc
φ1


f

=
αcc
φ2

m

=
αcl

m
=
−αcl φ0 
,
m
=
−αcl φ1 


m
=
−αcl φ2 

m

=
αcc

m
=
−αcc φ0 

m
=
−αcc
φ1 


m
φ2 
=
−αcc

f

=
αlX

m

=
αlX

f

=
βlX
m
=
βlX
θ̂ is distributed asymptotically normal as:
´
³ ¡
¢−1 ´
√ ³
> −1
n θ̂ − θ →d N 0, G V G
,
where G(θ) =
∂f (θ)
.
∂θ>
29
Table 1: Descriptive Statistics from the FES
Male
1968 to 2001
Female
Single
Married
Single
Married
Mean Std Dev Mean Std Dev Mean Std Dev Mean Std Dev
Age (22 to 65)
43.92
No hours dummy 0.28
Part time dummy 0.05
Full time dummy 0.68
Hourly wage
4.94
Total Expend.
118.37
Housing Expend. 39.57
Observations
10,958
Observed wages 7,663
13.39 48.60 13.51 50.41
0.45
0.19
0.40
0.43
0.22
0.04
0.20
0.16
0.47
0.77
0.42
0.41
2.43
4.74
2.27
3.86
99.10 192.51 129.48 99.53
41.62 59.61 62.93 39.07
31,871
12,967
25,208
7,271
30
13.31 46.51 13.60
0.49
0.35
0.48
0.37
0.25
0.43
0.49
0.40
0.49
1.96
3.33
1.66
75.63 192.51 129.48
37.91 59.61 62.93
31,871
20,291
Table 2: Unrestricted Estimates of the Sharing Rule.
φf0 cl
=
Πflym /Πfly
One Distribution Factor
Two Distribution Factors
−0.534
(0.056)
−1.707
(0.536)
−0.393
(0.089)
−0.327
(0.160)
2.297
(0.179)
5.796
(1.687)
1.714
(0.175)
1.635
(0.342)
−0.517
(0.055)
−1.634
(0.520)
−0.388
(0.088)
−0.329
(0.158)
2.300
(0.178)
5.794
(1.691)
1.765
(0.175)
1.726
(0.341)
−0.009
(0.003)
−0.038
(0.019)
−0.011
0.004
−0.015
(0.009)
+1
φf0 cc = Πfwlym /Πfwly + 1
m
= −Πm
φmcl
0
lym /Πly
m
φmcc
= −Πm
0
wlym /Πwly
φf1 cl = Πfz1 lym /Πfly
φf1 cc = Πfz1 wlym /Πfwly
m
φmcl
= −Πm
1
z1 lym /Πly
m
φmcc
= −Πm
1
z1 wlym /Πwly
***
***
***
**
***
***
***
***
φf2 cl = Πfz2 lym /Πfly
φf2 cc = Πfz2 wlym /Πfwly
m
φmcl
= −Πm
2
z2 lym /Πly
m
φmcc
= −Πm
2
z2 wlym /Πwly
Tests
φf = φm
φf cl = φf cc
φmcl = φmcc
φf cl = φmcl
φf cc = φmcc
df
6
2
2
2
2
χ2
19.49
5.80
1.05
8.65
6.09
p-value
0.003
0.055
0.592
0.013
0.048
df
9
3
3
3
3
χ2
21.73
6.27
1.92
8.26
5.81
***
***
***
**
***
***
***
***
***
**
***
**
p-value
0.010
0.099
0.590
0.041
0.121
´
³
f
Note: The sharing rule has the form: φ = φ0 + φ1 wfw+wm + φ2 (agem − agef ). Each sharing
rule parameter (φ0 , φ1 , and φ2 ) can be recovered from the restrictions on the reduced form estimates
in equation 9. Standard errors in parentheses. *, **, and *** indicate the coefficient is statistically
different from zero at the 10%, 5% and 1% significance levels, respectively
31
Table 3: Minimum Distance Sharing Rule Estimates
φ0
φ1
(1)
(2)
−0.317
(0.021)
1.584
(0.058)
−0.310
(0.021)
1.592
(0.057)
−0.004
(0.001)
0.485
(0.013)
φ2
φ0 + 12 φ1
0.475
(0.012)
Note: Standard errors in parentheses.
32
Table 4: Sharing Rule Estimates Under Alternative Measures of Public Goods
No Public Goods
φ0
φ1
φ0 + 21 φ1
0.008
(0.038)
0.773
(0.091)
0.395
(0.018)
Public Goods (i)
(housing)
−0.186
(0.028)
1.220
(0.073)
0.424
(0.015)
0%
17%
% of Total Consumption
φ0
φ1
φ0 + 21 φ1
Public Goods (ii)
(i + heat)
−0.239
(0.027)
1.352
(0.071)
0.437
(0.015)
% of Total Consumption
φ0
φ1
φ0 + 21 φ1
24%
Public Goods (iv)
(iii + transport)
−0.310
(0.013)
1.665
(0.038)
0.523
(0.010)
% of Total Consumption
47%
Note: Standard errors in parentheses.
33
Public Goods (iii)
(ii + durables)
−0.317
(0.015)
1.584
(0.013)
0.475
(0.010)
33%
Public Goods (v)
(iv + services)
−0.256
(0.008)
1.562
(0.025)
0.525
(0.007)
56%
Table 5: Minimum Distance Sharing Rule Estimates by
Birth Cohort.
Birth Cohort
1900 φ0
N : 3, 134
φ1
φ2
φ0 +
1910 φ0
N: 12,211
φ1
φ2
φ0 +
1920 φ0
N: 18,660
φ1
φ2
φ0 +
1930 φ0
N: 16,219
φ1
φ2
φ0 +
(1)
(2)
(3)
Birth Cohort
−0.099 −0.173 −0.260
(0.11) (0.09) (0.11)
0.470 0.778 1.008
(0.32) (0.25) (0.27)
−0.002
(0.007)
1
φ 0.136 0.216 0.244
2 1
(0.061) (0.050) (0.041)
−0.379 −0.347 −0.139
(0.053) (0.050) (0.054)
1.799 1.650 1.135
(0.154) (0.143) (0.122)
0.005
(0.003)
1
φ 0.520 0.478 0.428
2 1
(0.034) (0.033) (0.021)
−0.591 −0.568 −0.471
(0.058) (0.052) (0.030)
2.325 2.303 2.067
(0.171) (0.154) (0.056)
−0.006
(0.003)
1
φ 0.572 0.583 0.563
2 1
(0.036) (0.034) (0.010)
−0.195 −0.204 −0.154
(0.056) (0.041) (0.033)
1.248 1.330 1.071
(0.137) (0.122) (0.085)
−0.000
(0.003)
1
φ 0.429 0.461 0.382
2 1
(0.052) (0.031) (0.016)
1940 φ0
N: 15,284
φ1
φ2
φ0 +
1950 φ0
N: 11,692
φ1
φ2
φ0 +
1960 φ0
N: 7,974
φ1
φ2
φ0 +
1970 φ0
N: 2,585
φ1
φ2
φ0 +
(1)
(2)
(3)
−0.138 −0.152 −0.001
(0.066) (0.066) (0.052)
1.474 1.487 1.223
(0.154) (0.156) (0.117)
0.005
(0.004)
1
φ 0.599 0.591 0.611
2 1
(0.033) (0.033) (0.023)
−0.361 −0.348 −0.274
(0.093) (0.087) (0.078)
1.738 1.746 1.462
(0.240) (0.228) (0.191)
−0.004
(0.005)
1
φ 0.507 0.525 0.457
2 1
(0.048) (0.049) (0.033)
0.188 0.167 0.290
(0.153) (0.158) (0.169)
0.574 0.590 0.290
(0.332) (0.336) (0.370)
0.007
(0.008)
1
φ 0.475 0.462 0.435
2 1
(0.056) (0.056) (0.050)
−0.147 −0.193 0.017
(0.184) (0.264) (0.179)
0.997 1.210 0.703
(0.465) (0.613) (0.455)
0.295
(0.059)
1
φ 0.352 0.412 0.369
2 1
(0.089) (0.130) (0.098)
Note: Standard errors in parentheses. N indicates the sample size for each birth cohort. Column
3 contains estimates allowing for unobserved preference heterogeneity for leisure based on 100 random
draws. The covariance matrix of the reduced form estimates is based on the numerical Hessian for
column 1 and 2 and on the outer product of the gradient for column 3.
34
Table 6: Decomposition of the Change in Between and
Within Consumption Inequality.
Absolute Change
1978
2001
Change
Gini Index
Total Between
0.332
0.372
0.040
Mean Logarithmic Deviation
Total Between Within
0.285
0.337
0.052
0.197
0.255
0.057
0.137
0.204
0.067
0.060
0.050
−0.010
15.2
50.8
36.3
44.2
0.2
21.5
32.5
8.3
32.3
23.4
20.8
−1.0
17.1
16.3
9.4
38.9
27.1
32.1
0.0
16.4
23.1
15.8
78.1
48.4
98.3
5.7
12.3
63.3
Percentage of change attributable to:
Wages
18.2
Wage Sorting
51.7
Labor Supply
39.6
Labor Supply Sorting
40.4
Age Distribution
0.4
Household Composition
29.2
Age and Household
30.7
35
.7
.4
Gini Coefficient
.5
.6
Individual Earnings
Household Consumption
.3
Household Earnings
Adult Equivalent Consumption
1970
1980
1990
2000
Year
Figure 1: Trends in the Gini index for earnings and consumption.
0
Husband−wife earnings correlation
.1
.2
.3
Own calculations from the FES.
1970
1980
1990
2000
Year
Figure 2: Correlation in earnings across husbands and wives.
Own calculations from the FES.
36
.45
.4
.35
.3
.25
.2
1970
1980
1990
2000
Year
Female Share of Household Earnings
Potential Female Share
Figure 3: Fraction of actual household earnings provided by wife.
Source: Own calculation from the FES.
1968, Male, Married
1968, Female, Single
1968, Female, Married
1978, Male, Single
1978, Male, Married
1978, Female, Single
1978, Female, Married
1988, Male, Single
1988, Male, Married
1988, Female, Single
1988, Female, Married
1998, Male, Single
1998, Male, Married
1998, Female, Single
1998, Female, Married
0
.6
.4
.2
0
.6
.4
.2
0
Fraction
.2
.4
.6
0
.2
.4
.6
1968, Male, Single
0
20
40
0
20
40
0
20
40
0
Usual Weekly Hours
Graphs by year, Sex, and Marital Status
Figure 4: Histogram of usual weekly hours.
Source: Own calculation from the FES.
37
20
40
.36
.34
.36
.34
.26
.28
.3
.32
Gini Coefficient
.3
.32
.28
.26
1970
1980
1990
2000
Year
All Households
One and Two Adult Households
Figure 5: Consumption inequality for all households and one- and two-adult
households.
Source: UK National Statistics (1968–2001).
Mean Logarithmic Deviation
.25
.4
.3
.45
Gini Coefficient
.35
Total inequality
.2
Total inequality
.3
Between household
inequality
.25
.15
Between household
inequality
1970
1980
1990
2000
1970
1980
year
1990
2000
year
Figure 6: Total and between household decomposition of consumption inequality trends.
Source: Own calculation from the FES.
38
Mean Logarithmic Deviation, relative to 1968
.8
1
1.2
1.4
1.6
1.8
1970
1980
1990
2000
year
Total inequality
Within household inequality
Between household inequality
Figure 7: Relative changes in between and within household inequality
trends.
4
1
Source: Own calculation from the FES.
Cumulative Distribution
.4
.6
Kernel Density
2
3
.8
2001
1968
2001
0
0
.2
1
1968
0
.2
.4
.6
.8
Wife’s Share of Consumption
1
0
.2
.4
.6
.8
Wife’s Share of Consumption
1
Figure 8: Distribution of consumption inequality within households.
39
1
Lowest Rate/Top Rate
.4
.6
.8
.2
0
1970
1980
1990
year
Lowest Rate
Top Rate
Figure 9: Bottom and top marginal tax rates.
Source: UK National Statistics (1968–2001).
40
2000